\[ \int \frac {1}{x^2 \sqrt {c+d x^3} \left (4 c+d x^3\right )} \, dx \]
Optimal antiderivative \[ \frac {d^{\frac {1}{3}} \arctanh \left (\frac {c^{\frac {1}{6}} \left (c^{\frac {1}{3}}-2^{\frac {1}{3}} d^{\frac {1}{3}} x \right )}{\sqrt {d \,x^{3}+c}}\right ) 2^{\frac {1}{3}}}{24 c^{\frac {11}{6}}}-\frac {d^{\frac {1}{3}} \arctanh \left (\frac {\sqrt {d \,x^{3}+c}}{\sqrt {c}}\right ) 2^{\frac {1}{3}}}{72 c^{\frac {11}{6}}}+\frac {d^{\frac {1}{3}} \arctan \left (\frac {c^{\frac {1}{6}} \left (c^{\frac {1}{3}}+2^{\frac {1}{3}} d^{\frac {1}{3}} x \right ) \sqrt {3}}{\sqrt {d \,x^{3}+c}}\right ) 2^{\frac {1}{3}} \sqrt {3}}{72 c^{\frac {11}{6}}}-\frac {d^{\frac {1}{3}} \arctan \left (\frac {\sqrt {d \,x^{3}+c}\, \sqrt {3}}{3 \sqrt {c}}\right ) 2^{\frac {1}{3}} \sqrt {3}}{72 c^{\frac {11}{6}}}-\frac {\sqrt {d \,x^{3}+c}}{4 c^{2} x}+\frac {d^{\frac {1}{3}} \sqrt {d \,x^{3}+c}}{4 c^{2} \left (d^{\frac {1}{3}} x +c^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )}+\frac {d^{\frac {1}{3}} \left (c^{\frac {1}{3}}+d^{\frac {1}{3}} x \right ) \EllipticF \left (\frac {d^{\frac {1}{3}} x +c^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{d^{\frac {1}{3}} x +c^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \sqrt {\frac {c^{\frac {2}{3}}-c^{\frac {1}{3}} d^{\frac {1}{3}} x +d^{\frac {2}{3}} x^{2}}{\left (d^{\frac {1}{3}} x +c^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}\, 3^{\frac {3}{4}} \sqrt {2}}{12 c^{\frac {5}{3}} \sqrt {d \,x^{3}+c}\, \sqrt {\frac {c^{\frac {1}{3}} \left (c^{\frac {1}{3}}+d^{\frac {1}{3}} x \right )}{\left (d^{\frac {1}{3}} x +c^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}-\frac {3^{\frac {1}{4}} d^{\frac {1}{3}} \left (c^{\frac {1}{3}}+d^{\frac {1}{3}} x \right ) \EllipticE \left (\frac {d^{\frac {1}{3}} x +c^{\frac {1}{3}} \left (1-\sqrt {3}\right )}{d^{\frac {1}{3}} x +c^{\frac {1}{3}} \left (1+\sqrt {3}\right )}, i \sqrt {3}+2 i\right ) \left (\frac {\sqrt {6}}{2}-\frac {\sqrt {2}}{2}\right ) \sqrt {\frac {c^{\frac {2}{3}}-c^{\frac {1}{3}} d^{\frac {1}{3}} x +d^{\frac {2}{3}} x^{2}}{\left (d^{\frac {1}{3}} x +c^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}}{8 c^{\frac {5}{3}} \sqrt {d \,x^{3}+c}\, \sqrt {\frac {c^{\frac {1}{3}} \left (c^{\frac {1}{3}}+d^{\frac {1}{3}} x \right )}{\left (d^{\frac {1}{3}} x +c^{\frac {1}{3}} \left (1+\sqrt {3}\right )\right )^{2}}}} \]
command
integrate(1/x^2/(d*x^3+4*c)/(d*x^3+c)^(1/2),x, algorithm="fricas")
Fricas 1.3.8 (sbcl 2.2.11.debian) via sagemath 9.6 output
\[ \text {output too large to display} \]
Fricas 1.3.7 via sagemath 9.3 output \[ {\rm integral}\left (\frac {\sqrt {d x^{3} + c}}{d^{2} x^{8} + 5 \, c d x^{5} + 4 \, c^{2} x^{2}}, x\right ) \]